Integrand size = 35, antiderivative size = 295 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(b c C+3 b B d-4 a C d) x \sqrt {c+d x^2}}{3 b^2 d \sqrt {a+b x^2}}+\frac {C x^3 \sqrt {c+d x^2}}{3 b \sqrt {a+b x^2}}+\frac {\left (3 A b^2 d+8 a^2 C d-a b (c C+6 B d)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{5/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} (3 b B-4 a C) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/3*(3*B*b*d-4*C*a*d+C*b*c)*x*(d*x^2+c)^(1/2)/b^2/d/(b*x^2+a)^(1/2)+1/3*C* x^3*(d*x^2+c)^(1/2)/b/(b*x^2+a)^(1/2)+1/3*(3*A*b^2*d+8*a^2*C*d-a*b*(6*B*d+ C*c))*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d /b/c)^(1/2))/a^(1/2)/b^(5/2)/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^( 1/2)+1/3*a^(1/2)*(3*B*b-4*C*a)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1 /2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/( b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 4.40 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (c+d x^2\right ) \left (3 A b^2+a \left (-3 b B+4 a C+b C x^2\right )\right )+i c \left (3 A b^2 d+8 a^2 C d-a b (c C+6 B d)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (3 A b^2 d+4 a^2 C d-a b (c C+3 B d)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b^3 d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[c + d*x^2]*(A + B*x^2 + C*x^4))/(a + b*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*d*x*(c + d*x^2)*(3*A*b^2 + a*(-3*b*B + 4*a*C + b*C*x ^2)) + I*c*(3*A*b^2*d + 8*a^2*C*d - a*b*(c*C + 6*B*d))*Sqrt[1 + (b*x^2)/a] *Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c* (3*A*b^2*d + 4*a^2*C*d - a*b*(c*C + 3*B*d))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + ( d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b^3*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(616\) vs. \(2(295)=590\).
Time = 1.14 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A \sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}}+\frac {B x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}}+\frac {C x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} C \sqrt {a+b x^2} (b c-8 a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^3 \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 B \sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {4 c^{3/2} C \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b^2 \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {B c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {C x \sqrt {a+b x^2} (b c-8 a d)}{3 b^3 \sqrt {c+d x^2}}+\frac {2 B d x \sqrt {a+b x^2}}{b^2 \sqrt {c+d x^2}}+\frac {4 C x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b^2}-\frac {B x \sqrt {c+d x^2}}{b \sqrt {a+b x^2}}-\frac {C x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2}}\) |
Input:
Int[(Sqrt[c + d*x^2]*(A + B*x^2 + C*x^4))/(a + b*x^2)^(3/2),x]
Output:
(2*B*d*x*Sqrt[a + b*x^2])/(b^2*Sqrt[c + d*x^2]) + (C*(b*c - 8*a*d)*x*Sqrt[ a + b*x^2])/(3*b^3*Sqrt[c + d*x^2]) - (B*x*Sqrt[c + d*x^2])/(b*Sqrt[a + b* x^2]) - (C*x^3*Sqrt[c + d*x^2])/(b*Sqrt[a + b*x^2]) + (4*C*x*Sqrt[a + b*x^ 2]*Sqrt[c + d*x^2])/(3*b^2) + (A*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b] *x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b*x^2]*Sqrt[(a*( c + d*x^2))/(c*(a + b*x^2))]) - (2*B*Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*Ellip ticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b^2*Sqrt[(c*(a + b*x^ 2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*C*(b*c - 8*a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^3*Sqr t[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (B*c^(3/2)*S qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a *b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*c^( 3/2)*C*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a *d)])/(3*b^2*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2] )
Time = 9.25 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.67
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) \left (b^{2} A -a b B +a^{2} C \right ) x}{a \,b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {C x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{2}}+\frac {\left (\frac {b^{2} A d -a b B d +B \,b^{2} c +a^{2} C d -C a b c}{b^{3}}-\frac {\left (b^{2} A -a b B +a^{2} C \right ) \left (a d -b c \right )}{b^{3} a}-\frac {c \left (b^{2} A -a b B +a^{2} C \right )}{b^{2} a}-\frac {a c C}{3 b^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {B b d -C a d +C b c}{b^{2}}-\frac {\left (b^{2} A -a b B +a^{2} C \right ) d}{b^{2} a}-\frac {C \left (2 a d +2 b c \right )}{3 b^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(494\) |
| default | \(\frac {\sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \left (C \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{5}+3 A \sqrt {-\frac {b}{a}}\, b^{2} d^{2} x^{3}-3 B \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{3}+4 C \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{3}+C \sqrt {-\frac {b}{a}}\, a b c d \,x^{3}+3 A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c d -3 A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c d -3 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d +6 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d +4 C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d -C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}-8 C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d +C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}+3 A \sqrt {-\frac {b}{a}}\, b^{2} c d x -3 B \sqrt {-\frac {b}{a}}\, a b c d x +4 C \sqrt {-\frac {b}{a}}\, a^{2} c d x \right )}{3 b^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}\, d a}\) | \(652\) |
| risch | \(\frac {C x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b^{2}}+\frac {\left (-\frac {\left (3 B b d -5 C a d +C b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {\left (3 b^{2} A d -3 a b B d +3 B \,b^{2} c +3 a^{2} C d -4 C a b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{b \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 \left (b^{2} d A a -A \,b^{3} c -B \,a^{2} b d +B \,b^{2} c a +C \,a^{3} d -C \,a^{2} b c \right ) \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(667\) |
Input:
int((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOS E)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*((b*d*x^2+b*c) *(A*b^2-B*a*b+C*a^2)/a/b^3*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3*C/b^2*x*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+((A*b^2*d-B*a*b*d+B*b^2*c+C*a^2*d-C*a*b *c)/b^3-(A*b^2-B*a*b+C*a^2)/b^3*(a*d-b*c)/a-1/b^2*c*(A*b^2-B*a*b+C*a^2)/a- 1/3*a/b^2*c*C)/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a *d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2 ))-(1/b^2*(B*b*d-C*a*d+C*b*c)-(A*b^2-B*a*b+C*a^2)/b^2*d/a-1/3*C/b^2*(2*a*d +2*b*c))*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x ^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2) )-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
Time = 0.09 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (C a b^{2} c^{2} - {\left (8 \, C a^{2} b - 6 \, B a b^{2} + 3 \, A b^{3}\right )} c d\right )} x^{3} + {\left (C a^{2} b c^{2} - {\left (8 \, C a^{3} - 6 \, B a^{2} b + 3 \, A a b^{2}\right )} c d\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (C a b^{2} c^{2} - {\left (8 \, C a^{2} b - 6 \, B a b^{2} + 3 \, A b^{3}\right )} c d - {\left (4 \, C a^{2} b - 3 \, B a b^{2}\right )} d^{2}\right )} x^{3} + {\left (C a^{2} b c^{2} - {\left (8 \, C a^{3} - 6 \, B a^{2} b + 3 \, A a b^{2}\right )} c d - {\left (4 \, C a^{3} - 3 \, B a^{2} b\right )} d^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (C a b^{2} d^{2} x^{4} + C a^{2} b c d - {\left (8 \, C a^{3} - 6 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{2} + {\left (C a b^{2} c d - {\left (4 \, C a^{2} b - 3 \, B a b^{2}\right )} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a b^{4} d^{2} x^{3} + a^{2} b^{3} d^{2} x\right )}} \] Input:
integrate((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="fr icas")
Output:
-1/3*(((C*a*b^2*c^2 - (8*C*a^2*b - 6*B*a*b^2 + 3*A*b^3)*c*d)*x^3 + (C*a^2* b*c^2 - (8*C*a^3 - 6*B*a^2*b + 3*A*a*b^2)*c*d)*x)*sqrt(b*d)*sqrt(-c/d)*ell iptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((C*a*b^2*c^2 - (8*C*a^2*b - 6* B*a*b^2 + 3*A*b^3)*c*d - (4*C*a^2*b - 3*B*a*b^2)*d^2)*x^3 + (C*a^2*b*c^2 - (8*C*a^3 - 6*B*a^2*b + 3*A*a*b^2)*c*d - (4*C*a^3 - 3*B*a^2*b)*d^2)*x)*sqr t(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (C*a*b^2*d ^2*x^4 + C*a^2*b*c*d - (8*C*a^3 - 6*B*a^2*b + 3*A*a*b^2)*d^2 + (C*a*b^2*c* d - (4*C*a^2*b - 3*B*a*b^2)*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a* b^4*d^2*x^3 + a^2*b^3*d^2*x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x^{2}} \left (A + B x^{2} + C x^{4}\right )}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x**2+c)**(1/2)*(C*x**4+B*x**2+A)/(b*x**2+a)**(3/2),x)
Output:
Integral(sqrt(c + d*x**2)*(A + B*x**2 + C*x**4)/(a + b*x**2)**(3/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="ma xima")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(d*x^2 + c)/(b*x^2 + a)^(3/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="gi ac")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(d*x^2 + c)/(b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}\,\left (C\,x^4+B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(((c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + b*x^2)^(3/2),x)
Output:
int(((c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + b*x^2)^(3/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x)
Output:
(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d*x - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c**2*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d*x**3 + 3*sqr t(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*x - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2* c*x**4 + b**2*d*x**6),x)*a**3*c*d**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x **4 + b**2*d*x**6),x)*a**2*b**2*d**2 + 5*int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x **4 + b**2*d*x**6),x)*a**2*b*c**2*d - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x **2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x* *4 + b**2*d*x**6),x)*a**2*b*c*d**2*x**2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2* c*x**4 + b**2*d*x**6),x)*a*b**3*c*d + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x **2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x* *4 + b**2*d*x**6),x)*a*b**3*d**2*x**2 + 5*int((sqrt(c + d*x**2)*sqrt(a + b *x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c* x**4 + b**2*d*x**6),x)*a*b**2*c**2*d*x**2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b** 2*c*x**4 + b**2*d*x**6),x)*b**4*c*d*x**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c...